Fourier transformation of a class of hyperfunctions and formulation of the condition of local commutativity in the framework of localizable quantum field theory in terms of hyperfunctions

Theory of hyperfunctions is used for the formulation of aximatic local quantum field theory. Class $\mathscr L$ of hyperfunctions is introduced, which are local in coordinate as well as in the momentum space. Direct and inverse Fourier transformation of the hyperfunctions of the class $\mathscr L$ is defined and the relation of these hyperfunctions to generalized functions from the space $S_1{}^{1^\prime}$ is established. On the basis of notion of support of the hyper function the locality axiom is formulated for localizable field theories.